On the class of square Petrie matrices induced by cyclic permutations (Q1774781)

Let \(\sigma\) be a cyclic permutation of \(\{1,2,\dots,n+ 1\}\) and let \(A_\sigma\) be the \(n\times n\) Petrie matrix of \(\sigma\). In this note, it is shown that all such \(A_\sigma\) are similar over the field of two elements, all having characteristic polynomial \(\sum^n_{k=0} x^k\). These results do not hold over fields of characteristic other than 2.